Perfect Absorption and Strong Coupling in Supported MoS2 Multilayers

Perfect absorption and strong coupling are two highly sought-after regimes of light–matter interactions. Both regimes have been studied as separate phenomena in excitonic 2D materials, particularly in MoS2. However, the structures used to reach these regimes often require intricate nanofabrication. Here, we demonstrate the occurrence of perfect absorption and strong coupling in thin MoS2 multilayers supported by a glass substrate. We measure reflection spectra of mechanically exfoliated MoS2 flakes at various angles beyond the light-line via Fourier plane imaging and spectroscopy and find that absorption in MoS2 monolayers increases up to 74% at the C-exciton by illuminating at the critical angle. Perfect absorption is achieved for ultrathin MoS2 flakes (4–8 layers) with a notable angle and frequency sensitivity to the exact number of layers. By calculating zeros and poles of the scattering matrix in the complex frequency plane, we identify perfect absorption (zeros) and strong coupling (poles) conditions for thin (<10 layers) and thick (>10 layers) limits. Our findings reveal rich physics of light–matter interactions in bare MoS2 flakes, which could be useful for nanophotonic and light harvesting applications.

shows the schema of the setup. Light from a Laser driven light source (LDLS) is collimated. The collimated light passes through a polarizer (excitation is TE polarized) before entering an inverted microscope (Nikon Eclipse TE2000-e), depicted here as the vertical items. In the microscope, light is redirected to the objective (60×, 1.49 NA, Nikon CFI Apo TIRF 60XC Oil, MRD01691) with a 50/50 beamsplitter (BS, Chroma 21000). The beam size has to be large enough to fill the back aperture of the objective in order to have information at the highest angles. Note that due to the high NA and the filling of the back aperture, the polarization of the field is no longer preserved at the sample, as shown in the inset of Figure S2a.
The reflected light (orange arrows) is focused into a Nikon D300s digital color camera or on a fiber coupled spectrometer. The real image of the sample (bright field) is produced with the objective and a lens. Instead, by replacing such lens with a Bertrand lens it is possible to image the back focal plane of the objective (showed in dotted red, also called Fourier plane). Pictures of the Fourier plane are taken with the color digital camera Figure S1: a) Fourier plane spectroscopy setup. White light is collimated, polarized (TE) and then focused on the sample through an oil immersion objective. Due to the high NA of the objective, the polarization is no longer preserved at the illumination spot (see black lines in inset). The Fourier plane of the objective is focused using a Bertrand lens, positioned before the microscope's image plane. Then the spectra is taken with a fiber bundle mounted on an XYZ stage. The Fourier plane is imaged with a digital camera. b) Digital color picture of the Fourier plane of bare glass. The fiber bundle is shown as a series of bright spots in a line. This is the detection for spectra. c) Angular distribution in the Fourier plane. This shows also the angle ϕ, that determines the distribution of TE and TM components in the Fourier plane.
The Fourier plane of bare glass (picture taken with a digital camera) is shown in Figure  S1b. For angles above the glass/air critical angle θ > θ c = 41.5°there is total internal reflection. Therefore, in this region the lamp is perfectly reflected, thus we measured here the reference of the lamp for all angles. The fiber bundle was illuminated to make it visible in the same image. To cover all angles in the Fourier plane, the fiber bundle was mounted on a XYZ stage to control its position the Fourier plane.
The Fourier plane spreads the angular information along the radii as, r ∝ sin (θ) (shown in Figure S1c) and distributes the response to TE and TM polarized light. The intensity in the Fourier plane is given by The TE (s) and TM (p) polarized components of the electric field in the Fourier plane have been calculated before [1]. In this case, all the signal is parallel to the sample, therefore the field components expressions reduce to Where ϕ is the angle difference between the initial polarizer direction (polarization marked with a black line in Figure S1c) and the emitted light. From equation S2 it is clear that if ϕ = 0°, the field is purely TM polarized and that if ϕ = 90°the field is purely TE. All other angles in between have a mix of both components. Therefore, the TE polarized measurements were taken along ϕ = 90°and the TM ones along ϕ = 0°. Thus, the initial polarizer only rotates the orientation of the Fourier plane.

MoS Thickness characterization
Monolayers of MoS 2 were identified by measuring their Photoluminescence (PL) [2] as shown in Figure S2b. Photoluminescence was measured in an upright microscope (Nikon Eclipse LV150N) with a 100× objective (100× Nikon CFI60 TU Plan Epi ELWD). The broadband excitation lamp (CoolLED pE-300) was filtered with a bandpass of 400-488 nm. The measured power after the 100× objective was 49.2 mW. The collection spot was 1.8 µm in diameter. Once the monolayers were identified, the samples were characterized with Atomic Force Microscopy (AFM) and optical contrast. The monolayers were used as a baseline to characterize the height in both cases.
The thickness was also characterized by optical contrast [3]. The AFM measurements are challenging because the glass substrates are rough, so to avoid potential errors by AFM measurements we checked the thickness with a second method, optical contrast. Optical contrast is the difference of the intensity of light reflected by a flake and the substrate, 3 which is then normalized by the intensity reflected from a monolayer (identified by PL measurements as mentioned above). The normalized counts are shown in Figure S3. The analysis was done in ImageJ using profile analysis along the lines marked in colors on the bright field image in Figure S3a. We obtain the grey values (brightness of a pixel) by plotting the profile along the given line. We subtract the grey value of the background from the monolayers ( Figure S3c,e) and we use such value to normalize all other profiles. Therefore, we obtain the number of layers via the normalizing the contrast profiles, these are shown in Figure S3b-e. In general, both AFM and optical contrast agree. Figure S4 shows AFM measurements for the rest of the layers used in this study. Their thickness is shown in the caption. From a-c it shows the samples in the thin regime, discussed in section II in the main text. The thick flakes shown in S4e were used in section III in the main text. Most of the slabs in S4a-d were used for data presented in this supporting information.   C excitons to fit the experimental permittivity [4]:

Monolayer TM absorption
The absorption of the monolayer was described in the main text for TE-polarization. In Figure S6 we show the calculation of the TM-polarization absorption of a monolayer. Figure S6a shows the maximum of absorption (in the visible) for various angles. Interestingly, the minimum of reflection (in all the visible) reaches 100% at the critical angle. Meaning that at the critical angle, most of the light reflected by a monolayer is TM-polarized. Moreover in Figure S6b we can see that at θ c the absorption is 0 for all wavelengths (blue line). Thus, the TM light is perfectly reflected at θ c for all wavelengths. For TE, reflection at θ c reaches a maximum of 97% only for energies below the A-exciton ( Figure  2c). Regarding absorption, the maximum (C-exciton at 55 • ) only reaches 35%, thus the enhancement is of 1.34× at the C-exciton with respect to normal incidence.

Calculation of angular reflection spectra of TE-polarized light
As described in section III of the main text, thin MoS 2 layers absorb light in very different frequencies and angles. The difference is outstanding for a single-layer increase in thickness. In the main text ( Figure 3) we saw the different spectra for only 3 thicknesses. In Figure S7 we show the calculations of angular spectra for slabs with 1L -9L. The difference is clear on a single-layer level, thus making it an unambiguous technique to distinguish the thickness in thin slabs below 10L. Above that threshold all the angular spectra start reflecting most of the TE light and there is slight absorption only at the frequency of the excitons.

Experimental angular reflection spectra of TE-polarized light
The experimental equivalent of Figure S7 is shown in Figure S8. All the data presented in this figure is from different flakes than showed in Figure 3. Thus the angular spectra are reproducible for similar thickness flakes. In theory we showed that the thicknesses in panels c-f present perfect absorption. As mentioned above, noise in the experimental setup limits the measurable reflection to a minimum of 10 −2 . Figure S8c is clearly a 4L flake as shown in optical contrast ( Figure S3) and by comparing with theory ( Figure S7). Nevertheless the AFM measurements show a larger thickness. This is most likely related to the small size of the flake and to the big bubble in it ( Figure  S2c.iii).
Smaller differences between experiments and calculations may be given by the anisotropy in MoS 2 permittivity, which was not considered for the calculations (see Methods and SI

4.3Ŝ-matrix zeros in thin MoS 2 layers
The differences in reflection angular spectra by varying the thickness can be explained by finding the positions of the zeros of the S-matrix in the complex-ω plane. Figure S9ii shows the zeros for different angles beyond θ c . When the imaginary part of the frequency of the zero vanishes, they are fully real and there is perfect absorption. To see the correlation, Figure S9 shows the zeros positions in the complex frequency plane and the phase, φ. As was mentioned in the main text, φ, E ref /E inc = |r|e iφ , has a singularity when the reflected field is zero, E ref = 0. Moreover, beyond the critical angle T=0, thus the singularity appears when the absorption is perfect.
Due to the high loss of the excitons, the zeros of theŜ-matrix appear in the lower half of the complex plane for thin slabs ( Figure S9a), thus there are no singularities. Increasing the thickness pulls the zeros to the upper half of the complex plane. Due to its high oscillator strength (table 1), the first zeros to cross the real axis are the ones given by the C-exciton for 5L ( Figure S9b).
In general singularities come in pairs for thin slabs, as observed before for metasurfaces [6]. There is one pair per excitonic resonance ( Figure S9) and they have opposite topological charge. Even though they are related to the same resonance, the pairs appear at different angles and frequencies because the 3 zeros (one zero related to each exciton per angle) make a loop trajectory around the exciton when varying the angle of incidence ( Figure S9c). Thus they cross the real frequency axis at different angles and frequencies. All of them appear at different angles and frequencies. In Figure S9c we can see an example close to this situation, where for a 7L slab there are 5 singularities. Four of those are given by A and B excitons, the last one is given by the C-exciton.

Direct calculation of perfect absorption condition for thin films
The thickness and angle for which there will be perfect absorption of TE and TM-polarized light can be calculated from Fresnel equations. We consider an Attenuated total internal reflection situation as in the schema of Figure S10a. Here the incident media has a higher permittivity than the outgoing media, ε 1 > ε 3 , and the permittivity of the layer ε = (ε) + i (ε) is higher than both. TIR occurs for angles such that θ > θ c = arcsin n 3 n 1 . In this case, Where k 3 represents an evanescent wave and k 0 = ω/c. The reflection coefficient from the flake is given by r = r 12 + r 23 e 2ik 2 d 1 + r 12 r 23 e 2ik 2 d .
Then perfect absorption occurs when Using the Fresnel coefficients for TE waves we find that the condition for perfect absorption of TE-polarized light is This means that the thickness for which there is perfect absorption in the thin layer is For thin films we can consider k 2 d 1 and if ε 1,3 ε then k 1 k 2 k 2 2 . Then equation S8 can be approximated to By using that k 2 2 = k 2 0 ε + i ε − ε 1 sin 2 θ , equation S10 can be rewritten as This equation can be separated in real and imaginary part, giving rise to a pair of equations that must be fulfilled simultaneously to find perfect absorption.
This equations allow to obtain the thicknesses of slabs, frequencies and angles at which there will be perfect absorption for a material with a given permittivity in TIR configuration.
To find a connection between all possible angles and frequencies at which there could be perfect absorption, for arbitrary thicknesses, it is convenient to combine equations S12 and S13 by getting rid of d: From this equation, one can find the general relationship between the angle and the components of the permittivity, which for given ε(ω) yields the frequency dependence of the desired angles θ(ω). Then, substituting this dependence to the equation S12 or S13, one obtains the frequency dependence of the suitable thicknesses d(ω). The experimental limitation on the maximum observed angle on one side and the critical angle of total internal reflection on the other side gives a narrow window of allowable for the perfect absorption number of layers. Figure S10b shows the plots of equations S12 and S13 for MoS 2 for 7L. The intersection of both red lines (dashed and solid) are highlighted with circles and represent the points of perfect absorption in frequency and angles. Those points match to the singularities in the phase of the reflected wave ( Figure S10a). Figure S10c shows the same as S10b for thicknesses between 3L-9L (violet-red lines). The intersections of dashed and solid lines for a fixed thickness are marked with circles of corresponding color. Here is visible that intersections are only possible for 4L-8L in the studied spectral window (1.6 -2.9 eV). Before and after there are no intersections, and thus no perfect absorption. All the intersections of dashed and solid lines in Figures S10b,c lie on the black line which shows angles and frequencies for arbitrary thicknesses at which there could be perfect absorption (see equation S11). The frequency dependencies of the angles θ(ω) and thicknesses d(ω) obtained from the equations S14 and S13 are shown in Figure S10d.
The same approach can be used for TM-polarized light. Using the corresponding Fresnel coefficients, Then we find that the analogous of equation S9 for TM is (S16) 5 Reflection of thick slabs (d>10L)  Figure S11 shows the phase for the same thicknesses. The singularities mark precisely the points of perfect absorption. Their topological charge is also marked in grey C = 1 and black C = −1. Figure S12: Angle-dependent reflection spectra from thick MoS 2 slabs for various thicknesses around i) 32L, ii) 73L, iii) 84L, iv) 103L and v) 133L. a) Measurements of reflection of TE-polarized light for various increasing thicknesses (i-v). b) Calculation of the TEpolarized light reflection spectra matching the sale thicknesses. None of the thicknesses show perfect absorption. c) Calculation of the TM-polarized light reflection spectra for the same thicknesses. There is perfect absorption for all the thicknesses, giving rise to a singularity with topological charge shown in black C = −1 and in grey C = 1. Note the difference in the logarithmic scale of the colormap, whose minimum is 10 −6 . Figure S12b. The self-sustained optical mode below the light-line, due to the thickness of the slab, appears as a clear minimum in reflection in S12ii,iii. Increasing the thickness redshifts the mode until it starts interacting with the excitons as in Figure S12iv. Increasing the thickness even further causes a higher order optical mode to appear as in Figure S12v. TE-polarization does not show any points of perfect absorption in Figure S12b, but for the same thicknesses they appear for TM ( Figure S12c). Depending on the thickness 15 there can be up to 5 points of perfect absorption with opposite optical charges. The topological charge of each singularity is marked in black, C = −1 or grey C = 1.

Anisotropic MoS 2 calculation of TM-polarized reflected light and poles in thick layers
2D TMDs are highly anisotropic and MoS 2 is no exception [7]. Anisotropy is particularly important for TM-polarization because the electric field is not solely in-plane. To take anisotropy into account one needs to make a standard substitution in the reflection coefficients S15 and S5: k 2 → k 0 ε /ε ⊥ ε ⊥ − ε 1 sin 2 θ and ε 2 → ε . In Figure S14 the isotropic and anisotropic cases are compared. In this case n ⊥ = 2.74 [7] and ε was considered to be the same as given by the fitted Lorentzians in equation 3 in the main text, we also tried with the data in [7] and the changes are minimal (not showed here). Taking into account anisotropy provides a better correspondence with the experimental number of layers (see Figure 4 in the main text). Additionally, Figure S15a shows that the poles in the anisotropic case have the same behavior as the data presented in Figure 5. The B-exciton is strongly coupled above and below the light-line, whereas the A-exciton is not. Moreover, the Rabi splitting of the B-exciton with the photonic mode above the light-line is Ω R I = 0.12 eV and below the air light-line is Ω R I I = 0.16 eV, which is larger than the values presented in Figure 5 considering an isotropic flake.
Moreover, Figure S15b shows the singularities related to perfect absorption in the same flake. Therefore, the anisotropy does not change the general behavior described in the main text.   Figure S16 shows the same information for 97L for TE-polarized light.

Poles for TE-polarization thick slabs
Note that the three regions mentioned in the main text appear again. Region I (above the LL) is a two-port system because it can be illuminated from both sides of the flake. On the contrary, region II (beyond air LL) is limited to one-port because radiation happens solely through glass. In this region, perfect absorption can take place. In two-port systems, a similar concept to perfect absorption can be obtained, this is coherent perfect absorption where the phase has to be matched [8,9,10]. This increases the challenge of fabrication and design of the input of light in the samples.
In the TE-polarized light case, there are no perfect absorption points, since all the zeros are below the real axis.

Discussion about poles and zeros in TM-polarization
As mentioned in the main text, Figure 5d shows that two singularities can arise from the interaction with a single exciton because the zero is pulled towards the exciton (giving rise to a C = −1 topological charge) and then goes back to the upper half (giving rise to a C = 1 singularity) until the next exciton pulls it to the lower half of the complex plane. This gives rise to perfect absorption points because of the interplay between the photonic and the excitonic modes even if the coupling is not strong. This remark is interesting because, contrary to our case, phase singularities appearances have been related to strong coupling above the LL in organic molecules [11].
Also, the polaritonic eigenmodes do not perfectly absorb light. In order for them to happen a zero should occur at the same frequency and angle as a pole. It has been theorized before that the frequencies of the zeros and poles in a strongly coupled system, in a two-port symmetric system, would be the same only if the radiative losses of the cavity would be zero [9], which is not possible at all in a lossy slab. Actually, in such two-port systems, coherent perfect absorption and polaritons have been observed simultaneously only thanks to careful design and fabrication [12].